Diminished family: Difference between revisions

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== Diminished ==
== Diminished ==
{{see also|Jubilismic clan #Diminished}}
[[Subgroup]]: 2.3.5.7
 
Subgroup: 2.3.5.7


[[Comma list]]: 36/35, 50/49
[[Comma list]]: 36/35, 50/49
Line 25: Line 23:
{{Multival|legend=1|4 4 4 -3 -5 -2}}
{{Multival|legend=1|4 4 4 -3 -5 -2}}


[[POTE generator]]: ~3/2 = 699.523
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~3/2 = 699.523


{{Val list|legend=1| 4, 8d, 12 }}
{{Val list|legend=1| 4, 8d, 12 }}


[[Badness]]: 0.022401
[[Badness]]: 0.022401
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 36/35, 50/49, 56/55
Mapping: [{{val| 4 0 3 5 14 }}, {{val| 0 1 1 1 0 }}]
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109
Optimal GPV sequence: {{Val list| 4, 8d, 12, 32cddee, 44cddeee }}
Badness: 0.022132
Scales: [[diminished12]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 40/39, 50/49, 66/65
Mapping: [{{val| 4 0 3 5 14 15 }}, {{val| 0 1 1 1 0 0 }}]
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773
Optimal GPV sequence: {{Val list| 4, 8d, 12f, 20cdef }}
Badness: 0.019509
Scales: [[diminished12]]
=== Demolished ===
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 50/49
Mapping: [{{val| 4 0 3 5 -5 }}, {{val| 0 1 1 1 3 }}]
Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881
Optimal GPV sequence: {{Val list| 12, 28, 40de }}
Badness: 0.026574
=== Cohedim ===
This temperament has been documented in Graham Breed's temperament finder as ''hemidim'', the same name as [[Dimipent family #Hemidim|11-limit 4e & 24 and 13-limit 4ef & 24]]. For 11-limit 8bce & 12 temperament, ''cohedim'' arguably makes more sense.
Subgroup: 2.3.5.7.11
Comma list: 36/35, 50/49, 125/121
Mapping: [{{val| 4 1 4 6 6 }}, {{val| 0 2 2 2 3 }}]
Mapping generators: ~6/5, ~11/7
Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679
Optimal GPV sequence: {{Val list| 8bce, 12 }}
Badness: 0.054965
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 50/49, 66/65, 125/121
Mapping: [{{val| 4 1 4 6 6 7 }}, {{val| 0 2 2 2 3 3 }}]
Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299
Optimal GPV sequence: {{Val list| 8bcef, 12f }}
Badness: 0.041707


== Hemidim ==
== Hemidim ==

Revision as of 07:01, 18 March 2023

The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12EDO.

Dimipent

Subgroup: 2.3.5

Comma: 648/625

Mapping: [⟨4 0 3], ⟨0 1 1]]

POTE generator: ~3/2 = 699.507

Template:Val list

Badness: 0.047231

Diminished

Subgroup: 2.3.5.7

Comma list: 36/35, 50/49

Mapping: [⟨4 0 3 5], ⟨0 1 1 1]]

Wedgie: ⟨⟨ 4 4 4 -3 -5 -2 ]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 699.523

Template:Val list

Badness: 0.022401

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 56/55

Mapping: [⟨4 0 3 5 14], ⟨0 1 1 1 0]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 709.109

Optimal GPV sequence: Template:Val list

Badness: 0.022132

Scales: diminished12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 40/39, 50/49, 66/65

Mapping: [⟨4 0 3 5 14 15], ⟨0 1 1 1 0 0]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 713.773

Optimal GPV sequence: Template:Val list

Badness: 0.019509

Scales: diminished12

Demolished

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 50/49

Mapping: [⟨4 0 3 5 -5], ⟨0 1 1 1 3]]

Optimal tuning (POTE): ~6/5 = 1\4, ~3/2 = 689.881

Optimal GPV sequence: Template:Val list

Badness: 0.026574

Cohedim

This temperament has been documented in Graham Breed's temperament finder as hemidim, the same name as 11-limit 4e & 24 and 13-limit 4ef & 24. For 11-limit 8bce & 12 temperament, cohedim arguably makes more sense.

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 125/121

Mapping: [⟨4 1 4 6 6], ⟨0 2 2 2 3]]

Mapping generators: ~6/5, ~11/7

Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 101.679

Optimal GPV sequence: Template:Val list

Badness: 0.054965

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 50/49, 66/65, 125/121

Mapping: [⟨4 1 4 6 6 7], ⟨0 2 2 2 3 3]]

Optimal tuning (POTE): ~6/5 = 1\4, ~12/11 = 102.299

Optimal GPV sequence: Template:Val list

Badness: 0.041707

Hemidim

Subgroup: 2.3.5.7

Comma list: 49/48, 648/625

Mapping: [⟨4 0 3 8], ⟨0 2 2 1]]

Wedgie: ⟨⟨ 8 8 4 -6 -16 -13 ]]

POTE generator: ~7/6 = 252.555

Template:Val list

Badness: 0.086378

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 243/242

Mapping: [⟨4 0 3 8 -2], ⟨0 2 2 1 5]]

POTE generator: ~7/6 = 251.658

Optimal GPV sequence: Template:Val list

Badness: 0.056576

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 243/242

Mapping: [⟨4 0 3 8 -2 -1], ⟨0 2 2 1 5 5]]

POTE generator: ~7/6 = 252.225

Optimal GPV sequence: Template:Val list

Badness: 0.039030

Semidim

Subgroup: 2.3.5.7

Comma list: 245/243, 392/375

Mapping: [⟨8 0 6 -3], ⟨0 1 1 2]]

Wedgie: ⟨⟨ 8 8 16 -6 3 15 ]]

POTE generator: ~3/2 = 707.014

Optimal ET sequence: 8d, 24, 32c, 56c

Badness: 0.107523

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 245/243

Mapping: [⟨8 0 6 -3 15], ⟨0 1 1 2 1]]

POTE generator: ~3/2 = 706.645

Optimal GPV sequence: Template:Val list

Badness: 0.047598

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 507/500

Mapping: [⟨8 0 6 -3 15 17], ⟨0 1 1 2 1 1]]

POTE generator: ~3/2 = 707.376

Optimal GPV sequence: Template:Val list

Badness: 0.030597

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